Transitive Set of Ordinals is Ordinal
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Let $x$ be a transitive set of ordinals.
Then $x$ is itself an ordinal.
Let $x$ be a transitive set of ordinals according to the statement of the theorem.
We have from Class of All Ordinals is Well-Ordered by Subset Relation that $\On$ is well-ordered by $\subseteq$.
By Exists Ordinal Greater than Set of Ordinals there exists $\alpha$ such that $\alpha \notin x$.
Hence let $\alpha$ be the smallest ordinal not in $x$.
So all ordinals smaller than $\alpha$ are in $x$.
Hence from Ordinal equals its Initial Segment:
- $\alpha \subseteq x$
From Transitive Class of Ordinals is Subset of Ordinal not in it:
- $x \subseteq \alpha$
Hence by set equality:
- $x = \alpha$
Thus $x$ is an ordinal.
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 2$ Ordinals and transitivity: Theorem $2.2 \ (1)$